Transcript Linear Functions and Applications

Linear Functions and
Applications
Lesson 1.2
A Break Even Calculator
Consider this web site which helps a
business person know when they are
breaking even (starting to make money)
Note that the graph is a
line. Quite often, break
even analysis involves
a linear function.
2
Linear Function
A relationship f defined by
y  f ( x)  mx  b
for real numbers m and b is a
linear function
The independent variable is x
The dependent variable is y
3
Supply and Demand
Economists consider price to be
the independent variable
However
• They choose to plot price, p, on the vertical
axis
• Thus our text will consider p = f(q)
That is price is a function of quantity
Graph the function
(the calculator requires
that x be used, not q)
p  S (q)  1.4q  .6
4
Supply and Demand
The demand for an item can also be
represented by a linear function
• On the same set of axes, graph
p  D(q)  2q  3.2
p  S (q)  1.4q  .6
Note: we are only
interested in positive
values, Quadrant 1. Reset
the window with ♦E
5
Supply and Demand
Price
Set window for 0 < x < 3, 0 < y < 5
Supply
Demand
Quantity
Use the Trace feature (F3) to note values
of quantity and price
6
Supply and Demand
Price
What is the price and quantity where the
two functions are equal?
Supply
Demand
Intersection may
be found
symbolically or by
the calculator.
Quantity
This is called the point of equilibrium
7
Supply and Demand
Surplus is when excess supply exists
Supply
Demand
Surplus
Shortage
Shortage is when demand exceeds supply
8
Cost Analysis
Cost of manufacturing an item usually
consists of
• Fixed cost (rent, utilities, etc.)
• Cost per item (labor, materials, shipping …)
y  f ( x)  mx  b
This fits the description of a linear function
• The slope m is considered the "marginal cost"
• The y-intercept b is the fixed cost
9
Break Even Analysis
We compare Cost function with Revenue
Function
• Revenue is price times number sold
R( x)  p  x
Usually you must sell a certain number of
items to cover the fixed costs … beyond
that you are making a profit
• When R(x) > C(x)
• The break even point is when R(x) = C(x)
10
Break Even Analysis
Given
R( x)  4.95  x
C ( x)  525  2.15  x
Graph both and determine the point of
equilibrium
C(x)
loss
R(x)
11
Assignment
Lesson 1.2
Page 28
Exercises 1 – 25 odd,
29, 31, 37, 39
12